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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 11P: Additive number theory; partitions

Introduction

History

Applications and related fields

Subfields

• 11P05: Waring's problem and variants
• 11P21: Lattice points in specified regions
• 11P32: Goldbach-type theorems; other additive questions involving primes
• 11P55: Applications of the Hardy-Littlewood method, See also 11D85
• 11P70: Inverse problems of additive number theory [new in 2000]
• 11P81: Elementary theory of partitions, See also 05A17
• 11P82: Analytic theory of partitions
• 11P83: Partitions; congruences and congruential restrictions
• 11P99: None of the above but in this section

Parent field: 11: Number Theory

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Nathanson, Melvyn B.: "Additive number theory; The classical bases", Graduate Texts in Mathematics, 164. Springer-Verlag, New York, 1996. 342 pp. ISBN 0-387-94656-X

Number-theoretic properties of partitions are included in the extensive book by Andrews, George E.: "The theory of partitions", Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. 255 pp.

See also the references for number theory in general.

Software and tables

Other web sites with this focus

• Chen Shuwen has a web page covering the whole range of problems of equal sums of like powers; here is a similar site.

Selected topics at this site

• Asymptotic formula for number of partitions p(n).
• Listing all (restricted) partitions of n
• Number of partitions of n objects into subsets of bounded size
• How common are numbers expressible as a sum of 2 squares?
• How many integers less than n are the sum of two squares?
• How can we express a number as a sum of two squares (assuming that's possible!)
• What numbers are the sum of three squares? In how many ways?
• How many ways to write a number as a sum of 3 squares (Citation)
• Can every integer be represented as a sum of three squares? How? (Summary of other sums-of-squares questions too)
• How many representations of an integer as a sum of 3 squares?
• How hard to compute expressions of N as a sum of 4 squares?
• Enumerating representations of integers as sums of four squares
• Numbers expressible as sums of two cubes in multiple ways
• Solutions to x^3 + y^3 = z^3 + 1
• Which integers are the sum of three integer cubes? (unknown, e.g. n=33) [For sums of two cubes see elliptic curves]
• Is a number like 3 be expressible as a sum of three cubes in infinitely many ways?
• Any parametric families of integer solutions to x^3+y^3+z^3=1? (none known)
• Extensions of that fact about 1729.
• Smallest numbers representables as sums of 2 cubes in k ways.
• Multiple integer solutions to x^3+y^3=N (the taxicab example) and the elliptic curves y^2=x^3+D
• Pointer: Find sets of integers with equal sums of like powers.
• What is the ABC conjecture?
• Known cases of ABC conjecture; adjacent numbers with few divisors
• Extremal data when testing the ABC conjecture
• Generalizing the ABC Conjecture for integers (ABC Theorem for polynomials) to n summands
• Euler's conjecture generalizing Fermat's Last Theorem
• Can two squares sum to a fourth power? How about two consecutive squares?
• Solutions to x^5+y^5+z^5=w^5? (open)
• Can the sum of 2 fifth powers be a square? (Literature review of similar problems).
• Representing integers as sums of (fifth) powers of rational numbers
• Summary: how many N-th powers needed to sum to another N-th power?
• A continuum of problems linking Euler's conjecture to Waring's problem
• What is Waring's problem (write each N as a sum of powers)
• Summary of Waring's problem [Kevin Brown]
• Summary of status of Waring's problem
• Values of Waring's numbers G(N) and g(N) for low N.
• Citation for the case n=5 of Waring's problem (Chen: g(5)=37).
• Status of the Goldbach conjecture: open
• Goldbach conjecture
• Sylvester's Theorem: smallest integer not a positive linear combination of two given integers (a "postage stamp" problem).
• Frobenius problem -- largest total not expressible with n denominations of postage stamp
• Maximum integer not a linear combination of a few others (the postage stamp problem).
• Characterizing sets of integers by their difference sets.
• S-unit equation; finitude of solutions to Sum(a^{x_i})=power
• Counting (improper) q-adic representations of a number
• Finding subsets of a given set of integers whose sum is a multiple of N (and the Erdös-Ginzburg-Ziv theorem)